Exact Finite Local Sieve Structures for Symmetric Diagonal Affine Systems

This preprint develops an exact finite local sieve theory for symmetric diagonal affine systems. The paper studies a family of paired affine forms i + h_j + 1 and n - i - h_j + 1, and describes the finite residue classes that survive divisibility by a prescribed finite set of primes. This work continues the local obstruction framework introduced in the companion preprint Additive Encoding of Primality Constraints and Local Obstruction Structure in a Symmetric Affine System, but the present paper is self-contained. The main results include exact CRT-factorized counts for fixed-diagonal survivor fibers, two-variable packet-survivor spaces, finite diagonal-state distributions, and fiber moment laws. The paper also records the finite parity layer and explains the relation between the finite survivor structure and formal local singular factors. The results are finite congruence and local-sieve statements. No prime-distribution theorem, Goldbach-type theorem, or Hardy--Littlewood-type asymptotic is claimed. The purpose of the paper is to isolate and organize the finite local obstruction structure that precedes any analytic prime-value estimates.